Existence results for classes of infinite semipositone problems

  • Jerome GoddardII1,

    Affiliated with

    • Eun Kyoung Lee2,

      Affiliated with

      • Lakshmi Sankar3 and

        Affiliated with

        • R Shivaji4Email author

          Affiliated with

          Boundary Value Problems20132013:97

          DOI: 10.1186/1687-2770-2013-97

          Received: 23 October 2012

          Accepted: 5 April 2013

          Published: 19 April 2013

          Abstract

          We consider the problem

          { Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equa_HTML.gif

          where Δ p u = div ( | u | p 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, Ω is a smooth bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq6_HTML.gif, γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq7_HTML.gif and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq8_HTML.gif. Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq9_HTML.gif is presented.

          1 Introduction

          Consider the nonsingular boundary value problem:
          { Δ u = a u b u 2 c h ( x ) , x Ω , u = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ1_HTML.gif
          (1)

          where Ω is a smooth bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq10_HTML.gif, Δ u = div ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq11_HTML.gif is the Laplacian of u and h : Ω ¯ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq12_HTML.gif is a C 1 ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq13_HTML.gif function satisfying h ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq14_HTML.gif for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq15_HTML.gif, h ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq16_HTML.gif, max x Ω ¯ h ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq17_HTML.gif and h ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq18_HTML.gif for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq19_HTML.gif. Existence of positive solutions of problem (1) was studied in [1]. In particular, it was proved that given an a > λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq20_HTML.gif and b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif there exists a c ( a , b , Ω ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq21_HTML.gif such that for c < c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq22_HTML.gif (1) has positive solutions. Here, λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq23_HTML.gif is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when a λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq24_HTML.gif. Later in [2], these results were extended to the case of the p-Laplacian operator, Δ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq25_HTML.gif, where Δ p u = div ( | u | p 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif. Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity f ( s , x ) = a s b s 2 c h ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq26_HTML.gif satisfies f ( 0 , x ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq27_HTML.gif for some x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq28_HTML.gif. See [39] for some existence results for semipositone problems.

          In this paper, we study positive solutions to the singular boundary value problem:
          { Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ2_HTML.gif
          (2)

          where Δ p u = div ( | u | p 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq1_HTML.gif, p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, Ω is a smooth bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq10_HTML.gif, α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, and γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif. In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity f ( s ) = a s p 1 b s γ 1 c s α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq31_HTML.gif satisfies lim s 0 + f ( s ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq32_HTML.gif. One can refer to [1014], and [1517] for some recent existence results of infinite semipositone problems. We establish the following theorem.

          Theorem 1.1 Given a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq33_HTML.gif, γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq34_HTML.gif, and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, there exists a constant c 1 = c 1 ( a , b , α , p , γ , Ω ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq35_HTML.gif such that for c < c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif, (2) has a positive solution.

          Remark 1.1 In the nonsingular case ( α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq37_HTML.gif), positive solutions exist only when a > λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq20_HTML.gif (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif.

          Next, we study positive radial solutions to the problem:
          { Δ p u = K ( | x | ) ( a u p 1 b u γ 1 c u α ) , x Ω , u = 0 , if  | x | = r 0 , u 0 , as  | x | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ3_HTML.gif
          (3)
          where Ω = { x R n | | x | > r 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq38_HTML.gif is an exterior domain, n > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq39_HTML.gif, a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif, b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq5_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq40_HTML.gif, α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq2_HTML.gif, γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif and K : [ r 0 , ) ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq41_HTML.gif belongs to a class of continuous functions such that lim r K ( r ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq42_HTML.gif. By using the transformation: r = | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq43_HTML.gif and s = ( r r 0 ) n + p p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq44_HTML.gif, we reduce (3) to the following boundary value problem:
          { ( | u | p 2 u ) = h ( s ) ( a u p 1 b u γ 1 c u α ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ4_HTML.gif
          (4)

          where h ( s ) = ( p 1 n p ) p r 0 p s p ( n 1 ) n p K ( r 0 s ( p 1 ) n p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq45_HTML.gif. We assume:

          ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif) K C ( [ r 0 , ) , ( 0 , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq47_HTML.gif and satisfies K ( r ) < 1 r n + θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq48_HTML.gif for r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq49_HTML.gif, and for some θ such that ( n p p 1 ) α < θ < n p p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq50_HTML.gif.

          With the condition ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif), h satisfies:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ5_HTML.gif
          (5)

          We note that if θ n p p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq51_HTML.gif then h ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq52_HTML.gif is nonsingular at 0 and h C ( [ 0 , 1 ] , ( 0 , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq53_HTML.gif. In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when θ < n p p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq54_HTML.gif in which, h may be singular at 0. Note that in this case h ˆ = inf s ( 0 , 1 ) h ( s ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq55_HTML.gif.

          Remark 1.2 Note that ρ + α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq56_HTML.gif since θ > ( n p p 1 ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq57_HTML.gif.

          We then establish the following theorem.

          Theorem 1.2 Given a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq58_HTML.gif, γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq34_HTML.gif, α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, and assume ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq46_HTML.gif) holds. Then there exists a constant c 2 = c 2 ( a , b , α , p , γ ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq59_HTML.gif such that for c < c 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq60_HTML.gif, (3) has a positive radial solution.

          Finally, we prove a bifurcation result for the problem
          { Δ p u = a u p 1 b u γ 1 u α , x Ω , u = 0 , on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ6_HTML.gif
          (6)

          where Ω is a smooth bounded domain in R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq3_HTML.gif, a is a positive parameter, b , α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq61_HTML.gif, p > 1 + α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq62_HTML.gif and γ > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq30_HTML.gif. We prove the following.

          Theorem 1.3 The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq63_HTML.gif at ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq64_HTML.gif (as shown in Figure 1).
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig1_HTML.jpg
          Figure 1

          Bifurcation diagram, a vs. u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq65_HTML.gif for ( 6 ).

          Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function ψ W 1 , p ( Ω ) C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq66_HTML.gif that satisfies
          { Ω | ψ | p 2 ψ w d x Ω a ψ p 1 b ψ γ 1 c ψ α w d x , for every  w W , ψ > 0 , in  Ω , ψ = 0 , on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equb_HTML.gif
          and by a supersolution we mean a function Z W 1 , p ( Ω ) C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq67_HTML.gif that satisfies:
          { Ω | Z | p 2 Z w d x Ω a Z p 1 b Z γ 1 c Z α w d x , for every  w W , Z > 0 , in  Ω , Z = 0 , on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equc_HTML.gif

          where W = { ξ C 0 ( Ω ) | ξ 0  in  Ω } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq68_HTML.gif. The following lemma was established in [13].

          Lemma 1.4 (see [13, 18])

          Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that ψ Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq69_HTML.gif in Ω. Then (2) has a solution u such that ψ u Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq70_HTML.gif in Ω.

          Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that lim x Ω Δ p ψ = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq71_HTML.gif and Δ p ψ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq72_HTML.gif in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form ψ = k ϕ 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif, where k is an appropriate positive constant, β ( 1 , p p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq74_HTML.gif and ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif is the eigenfunction corresponding to the first eigenvalue of Δ p ϕ = λ | ϕ | p 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq76_HTML.gif in Ω, ϕ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq77_HTML.gif on Ω.

          In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when Ω = ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq78_HTML.gif and p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq79_HTML.gif.

          2 Proof of Theorem 1.1

          We first construct a subsolution. Consider the eigenvalue problem Δ p ϕ = λ | ϕ | p 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq80_HTML.gif in Ω, ϕ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq77_HTML.gif on Ω. Let ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be an eigenfunction corresponding to the first eigenvalue λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq23_HTML.gif such that ϕ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq81_HTML.gif and ϕ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq82_HTML.gif. Also, let δ , m , μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq83_HTML.gif be such that | ϕ 1 | m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq84_HTML.gif in Ω δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq85_HTML.gif and ϕ 1 μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq86_HTML.gif in Ω Ω δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq87_HTML.gif, where Ω δ = { x Ω | d ( x , Ω ) δ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq88_HTML.gif. Let β ( 1 , p p 1 + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq89_HTML.gif be fixed. Here, note that since α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif, p p 1 + α > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq90_HTML.gif. Choose a k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that 2 b k γ p + β p 1 λ 1 k α a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq92_HTML.gif. Define c 1 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq93_HTML.gif. Note that c 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq94_HTML.gif by the choice of k and β. Let ψ = k ϕ 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq95_HTML.gif. Then
          Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equd_HTML.gif
          To prove ψ is a subsolution, we need to establish:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ7_HTML.gif
          (7)
          in Ω if c < c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif. To achieve this, we split the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq96_HTML.gif into three, namely,
          k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Eque_HTML.gif
          Now to prove (7) holds in Ω, it is enough to show the following three inequalities:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ8_HTML.gif
          (8)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ9_HTML.gif
          (9)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ10_HTML.gif
          (10)
          From the choice of k, ( a β p 1 λ 1 k α ) 2 b k γ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq97_HTML.gif, hence,
          1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ11_HTML.gif
          (11)
          Using ϕ 1 μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq98_HTML.gif in Ω Ω δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq87_HTML.gif and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq99_HTML.gif
          1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 ) 2 k α ϕ 1 α β c k α ϕ 1 α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ12_HTML.gif
          (12)
          Finally, since | ϕ 1 | m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq100_HTML.gif, in Ω δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq85_HTML.gif, and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq101_HTML.gif,
          k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 p β ( p 1 ) α β c k α ϕ 1 α β ϕ 1 p β ( p 1 + α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equf_HTML.gif
          Since p β ( p 1 + α ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq102_HTML.gif,
          k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c k α ϕ 1 α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ13_HTML.gif
          (13)

          From (11), (12) and (13) we see that equation (7) holds in Ω, if c < c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq36_HTML.gif. Next, we construct a supersolution. Let e be the solution of Δ p e = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq103_HTML.gif in Ω , e = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq104_HTML.gif on Ω. Choose M ¯ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq105_HTML.gif such that a u p 1 b u γ 1 c u α M ¯ p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq106_HTML.gif u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq107_HTML.gif and M ¯ e ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq108_HTML.gif. Define Z = M ¯ e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq109_HTML.gif. Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

          3 Proof of Theorem 1.2

          We begin the proof by constructing a subsolution. Consider
          ( | ϕ | p 2 ϕ ) = λ | ϕ | p 2 ϕ , t ( 0 , 1 ) , ϕ ( 0 ) = ϕ ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ14_HTML.gif
          (14)
          Let ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be an eigenfunction corresponding to the first eigenvalue of (14) such that ϕ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq81_HTML.gif and ϕ 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq110_HTML.gif. Then there exist d 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq111_HTML.gif such that 0 < ϕ 1 ( t ) d 1 t ( 1 t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq112_HTML.gif for t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq113_HTML.gif. Also, let ϵ < ϵ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq114_HTML.gif and m , μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq115_HTML.gif be such that | ϕ 1 | m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq116_HTML.gif in ( 0 , ϵ ] [ 1 ϵ , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq117_HTML.gif and ϕ 1 μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq86_HTML.gif in ( ϵ , 1 ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq118_HTML.gif. Let β ( 1 , p ρ p 1 + α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq119_HTML.gif be fixed and choose k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that 2 b k γ p + β p 1 λ 1 k α h ˆ a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq120_HTML.gif. Define c 2 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq121_HTML.gif. Then c 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq122_HTML.gif by the choice of k and β. Let ψ = k ϕ 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif. This implies that:
          ( | ψ | p 2 ψ ) = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equg_HTML.gif
          To prove ψ is a subsolution, we need to establish:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ15_HTML.gif
          (15)
          Here, we note that the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = h ˆ k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) h ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq123_HTML.gif h ( t ) ( a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq124_HTML.gif, where h ˆ = inf s ( 0 , 1 ) h ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq125_HTML.gif. Now to prove (15) holds in ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq126_HTML.gif, it is enough to show the following three inequalities:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ16_HTML.gif
          (16)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ17_HTML.gif
          (17)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ18_HTML.gif
          (18)
          From the choice of k, ( a β p 1 λ 1 k α h ˆ ) 2 b k γ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq127_HTML.gif, hence,
          1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ19_HTML.gif
          (19)
          Using ϕ 1 μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq98_HTML.gif in ( ϵ , 1 ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq118_HTML.gif and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq128_HTML.gif
          1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 h ˆ ) 2 k α ϕ 1 α β c k α ϕ 1 α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ20_HTML.gif
          (20)
          Next, we prove (18) holds in ( 0 , ϵ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq129_HTML.gif. Since | ϕ 1 | m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq130_HTML.gif, and p β ( p 1 ) > α β + ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq131_HTML.gif
          k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 ρ k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β d 1 ρ t ρ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equh_HTML.gif
          Since h ( t ) 1 t ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq132_HTML.gif in ( 0 , ϵ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq129_HTML.gif, and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq133_HTML.gif,
          k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c h ( t ) k α ϕ 1 α β . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ21_HTML.gif
          (21)

          Proving (18) holds in [ 1 ϵ , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq134_HTML.gif is straightforward since h is not singular at t = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq135_HTML.gif. Thus, from equations (19), (20) and (21), we see that (15) holds in ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq126_HTML.gif. Hence, ψ is a subsolution. Let Z = M ¯ e http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq109_HTML.gif where e satisfies ( | e | p 2 e ) = h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq136_HTML.gif in ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq137_HTML.gif, e ( 0 ) = e ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq138_HTML.gif and M ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq139_HTML.gif is such that a u p 1 b u γ 1 c u α M ¯ p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq140_HTML.gif u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq107_HTML.gif and M ¯ e ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq108_HTML.gif. Then Z is a supersolution of (4) and there exists a solution u of (4) such that u [ ψ , Z ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq141_HTML.gif. Thus, Theorem 1.2 is proven.

          4 Proof of Theorem 1.3

          We first prove (6) has a positive solution for every a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif. We begin by constructing a subsolution. Let ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq75_HTML.gif be as in the proof of Theorem 1.1 (see Section 2). Let β ( 1 , p p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq74_HTML.gif, and choose a k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq91_HTML.gif such that b k γ p + β p 1 λ 1 k α a http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq142_HTML.gif. Let ψ = k ϕ 1 β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq73_HTML.gif. Then
          Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equi_HTML.gif
          To prove ψ is a subsolution, we will establish:
          k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) a k p 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ22_HTML.gif
          (22)
          in Ω. To achieve this, we rewrite the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq96_HTML.gif as k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq143_HTML.gif. Now to prove (22) holds in Ω, it is enough to show k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( γ 1 α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq144_HTML.gif. From the choice of k, ( a β p 1 λ 1 k α ) b k γ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq145_HTML.gif, hence,
          k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equj_HTML.gif

          Thus, ψ is a subsolution. It is easy to see that Z = ( a b ) 1 γ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq146_HTML.gif is a supersolution of (6). Since k, can be chosen small enough, ψ Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq69_HTML.gif. Thus, (6) has a positive solution for every a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq4_HTML.gif. Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq147_HTML.gif is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq63_HTML.gif at ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq64_HTML.gif.

          5 Numerical results

          Consider the boundary value problem
          { u ( x ) = a u b u 2 c u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ23_HTML.gif
          (23)
          where a , b > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq58_HTML.gif, c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq40_HTML.gif and α ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq29_HTML.gif. Using the quadrature method (see [19]), the bifurcation diagram of positive solutions of (23) is given by
          G ( ρ , c ) = 0 ρ d s [ 2 ( F ( ρ ) F ( s ) ) ] = 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ24_HTML.gif
          (24)
          where F ( s ) : = 0 s f ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq148_HTML.gif where f ( t ) = a t b t 2 c t α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq149_HTML.gif and ρ = u ( 1 2 ) = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq150_HTML.gif. We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when a = 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq151_HTML.gif ( < λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq152_HTML.gif) and b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif for different values of α.
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig2_HTML.jpg
          Figure 2

          Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq154_HTML.gif , b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

          Bifurcation diagrams of positive solutions of (23) when a = 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq156_HTML.gif ( > λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq157_HTML.gif) and b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif for different values of α is shown in Figure 3.
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig3_HTML.jpg
          Figure 3

          Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 15 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq158_HTML.gif , b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

          Finally, we provide the exact bifurcation diagram for (6) when p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq79_HTML.gif, and Ω = ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq78_HTML.gif. Consider
          { u ( x ) = a u b u 2 u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ25_HTML.gif
          (25)
          where a , b , α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq159_HTML.gif. The bifurcation diagram of positive solutions of (25) is given by
          G ˜ ( ρ , a ) = 0 ρ d s [ 2 ( F ˜ ( ρ ) F ˜ ( s ) ) ] = 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ26_HTML.gif
          (26)
          where F ˜ ( s ) : = 0 s f ˜ ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq160_HTML.gif where f ˜ ( t ) = a t b t 2 t α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq161_HTML.gif and ρ = u ( 1 2 ) = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq150_HTML.gif. The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when α = 0.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq162_HTML.gif and b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq153_HTML.gif.
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig4_HTML.jpg
          Figure 4

          Bifurcation diagram, a vs. ρ for ( 25 ) with α = 0.5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq163_HTML.gif , b = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_IEq155_HTML.gif .

          Declarations

          Acknowledgements

          EK Lee was supported by 2-year Research Grant of Pusan National University.

          Authors’ Affiliations

          (1)
          Department of Mathematics, Auburn University Montgomery
          (2)
          Department of Mathematics Education, Pusan National University
          (3)
          Department of Mathematics & Statistics, Mississippi State University
          (4)
          Department of Mathematics & Statistics, University of North Carolina at Greensboro

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          © Goddard II et al.; licensee Springer. 2013

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